We report theoretical and experimental studies on generation of high-order optical vortex beams using fiber-based nanostructured gradient index converters. The components were made of two types of in-house developed soft glasses with lead, bismuth, gallium compositions to ensure the high refractive contrast of the synthesized glasses. As a result, the vortex components, fabricated using the stack-and-draw method, allow converting Gaussian beam into vortex beams with high-order topological charges in the visible and near-infrared regions. We experimentally verified the formation of optical vortex beams with topological charge up to 5 by using the free-space binary-nanostructured vortex components. In addition, the vortex converter integrated into the standard fiber tip generated optical vortex beams with topological charge up to 4.